Optimal. Leaf size=27 \[ -\frac{c^2}{a^2 x}+\frac{2 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.121187, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 43} \[ -\frac{c^2}{a^2 x}+\frac{2 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx\\ &=\frac{c^2 \int \frac{e^{4 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \frac{(1+a x)^2}{x^2} \, dx}{a^2}\\ &=\frac{c^2 \int \left (a^2+\frac{1}{x^2}+\frac{2 a}{x}\right ) \, dx}{a^2}\\ &=-\frac{c^2}{a^2 x}+c^2 x+\frac{2 c^2 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.103305, size = 29, normalized size = 1.07 \[ -\frac{c^2}{a^2 x}+\frac{2 c^2 \log (a x)}{a}+c^2 x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.045, size = 28, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{{a}^{2}x}}+x{c}^{2}+2\,{\frac{{c}^{2}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0413, size = 36, normalized size = 1.33 \begin{align*} c^{2} x + \frac{2 \, c^{2} \log \left (x\right )}{a} - \frac{c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56775, size = 65, normalized size = 2.41 \begin{align*} \frac{a^{2} c^{2} x^{2} + 2 \, a c^{2} x \log \left (x\right ) - c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.292655, size = 26, normalized size = 0.96 \begin{align*} \frac{a^{2} c^{2} x + 2 a c^{2} \log{\left (x \right )} - \frac{c^{2}}{x}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12239, size = 127, normalized size = 4.7 \begin{align*} -\frac{2 \, c^{2} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{2} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{c^{2} + \frac{2 \, c^{2}}{a x - 1}}{a^{2}{\left (\frac{1}{{\left (a x - 1\right )} a} + \frac{1}{{\left (a x - 1\right )}^{2} a}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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